Data.Number.Erf:$dmerfcx from erf-2.0.0.0

Percentage Accurate: 100.0% → 100.0%

Time: 37.0s

Alternatives: 16

Speedup: 1.0×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (exp (* y y))))

C

double code(double x, double y) {
	return x * exp((y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * exp((y * y))
end function

Java

public static double code(double x, double y) {
	return x * Math.exp((y * y));
}

Python

def code(x, y):
	return x * math.exp((y * y))

Julia

function code(x, y)
	return Float64(x * exp(Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * exp((y * y));
end

Wolfram

code[x_, y_] := N[(x * N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (exp (* y y))))

C

double code(double x, double y) {
	return x * exp((y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * exp((y * y))
end function

Java

public static double code(double x, double y) {
	return x * Math.exp((y * y));
}

Python

def code(x, y):
	return x * math.exp((y * y))

Julia

function code(x, y)
	return Float64(x * exp(Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * exp((y * y));
end

Wolfram

code[x_, y_] := N[(x * N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (exp (* y y))))

C

double code(double x, double y) {
	return x * exp((y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * exp((y * y))
end function

Java

public static double code(double x, double y) {
	return x * Math.exp((y * y));
}

Python

def code(x, y):
	return x * math.exp((y * y))

Julia

function code(x, y)
	return Float64(x * exp(Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * exp((y * y));
end

Wolfram

code[x_, y_] := N[(x * N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (exp y)))

C

double code(double x, double y) {
	return x * exp(y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * exp(y)
end function

Java

public static double code(double x, double y) {
	return x * Math.exp(y);
}

Python

def code(x, y):
	return x * math.exp(y)

Julia

function code(x, y)
	return Float64(x * exp(y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * exp(y);
end

Wolfram

code[x_, y_] := N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(y \cdot 1\right)}} \]

   2. metadata-eval N/A

      \[\leadsto x \cdot e^{y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right)} \]

   3. metadata-eval N/A

      \[\leadsto x \cdot e^{y \cdot \left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto x \cdot e^{y \cdot \left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right)} \]

   5. distribute-lft-out N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)}} \]

   6. div-inv N/A

      \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right)} \]

   7. div-inv N/A

      \[\leadsto x \cdot e^{y \cdot \left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right)} \]

   8. flip-+ N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}}} \]

   9. +-inverses N/A

      \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   11. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   12. *-rgt-identity N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot 1\right)} \cdot 0}{0}} \]

   13. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

   14. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

   15. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

   16. distribute-lft-out N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

   17. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   19. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

   20. difference-of-squares N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

   21. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\color{blue}{\frac{y}{2} - \frac{y}{2}}}} \]

   22. flip-+ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y}{2} + \frac{y}{2}}} \]

   23. count-2 N/A

       \[\leadsto x \cdot e^{\color{blue}{2 \cdot \frac{y}{2}}} \]

4. Applied rewrites 71.4%

   \[\leadsto x \cdot e^{\color{blue}{y}} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 0.2)
   (* x (fma y (fma y (* (* y y) 0.5) y) 1.0))
   (* 0.16666666666666666 (* x (* y (* y (* y (* y (* y y)))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.2) {
		tmp = x * fma(y, fma(y, ((y * y) * 0.5), y), 1.0);
	} else {
		tmp = 0.16666666666666666 * (x * (y * (y * (y * (y * (y * y))))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.2)
		tmp = Float64(x * fma(y, fma(y, Float64(Float64(y * y) * 0.5), y), 1.0));
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(y * Float64(y * Float64(y * Float64(y * Float64(y * y)))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.2], N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(x * N[(y * N[(y * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.20000000000000001

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right) + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right) + 1\right) \]

      3. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)} + 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right), 1\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{2} + 1\right)}, 1\right) \]

      6. distribute-lft-in N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + y \cdot 1}, 1\right) \]

      7. *-rgt-identity N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + \color{blue}{y}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} \cdot {y}^{2}, y\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} \cdot \frac{1}{2}}, y\right), 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} \cdot \frac{1}{2}}, y\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{2}, y\right), 1\right) \]

      12. lower-*.f64 99.5

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot 0.5, y\right), 1\right) \]

   5. Applied rewrites 99.5%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)} \]

   if 0.20000000000000001 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right) \]

      3. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} + 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right), 1\right)} \]

   5. Applied rewrites 83.4%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right), y\right), 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot \left(y \cdot \left(y \cdot \left(y \cdot \frac{1}{6}\right) + \frac{1}{2}\right)\right) + y, 1\right) \]

      2. lift-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)} + \frac{1}{2}\right)\right) + y, 1\right) \]

      3. lift-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)}\right) + y, 1\right) \]

      4. lift-*.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right)} + y, 1\right) \]

      5. flip-+ N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{\left(\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right)\right) - y \cdot y}{\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right) - y}}, 1\right) \]

      6. clear-num N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{\frac{\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right) - y}{\left(\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right)\right) - y \cdot y}}}, 1\right) \]

      7. lower-/.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{\frac{\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right) - y}{\left(\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right)\right) - y \cdot y}}}, 1\right) \]

      8. clear-num N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{\color{blue}{\frac{1}{\frac{\left(\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right)\right) - y \cdot y}{\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right) - y}}}}, 1\right) \]

      9. flip-+ N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{\frac{1}{\color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \frac{1}{6}, \frac{1}{2}\right)\right) + y}}}, 1\right) \]

   7. Applied rewrites 83.4%

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right), y \cdot \left(y \cdot y\right), y\right)}}}, 1\right) \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(x \cdot {y}^{6}\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{6} \cdot \left(x \cdot {y}^{\color{blue}{\left(5 + 1\right)}}\right) \]

      4. pow-plus N/A

         \[\leadsto \frac{1}{6} \cdot \left(x \cdot \color{blue}{\left({y}^{5} \cdot y\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{6} \cdot \left(x \cdot \color{blue}{\left(y \cdot {y}^{5}\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{1}{6} \cdot \left(x \cdot \color{blue}{\left(y \cdot {y}^{5}\right)}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot {y}^{\color{blue}{\left(4 + 1\right)}}\right)\right) \]

      8. pow-plus N/A

         \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \color{blue}{\left({y}^{4} \cdot y\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(y \cdot {y}^{4}\right)}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \color{blue}{\left(y \cdot {y}^{4}\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(y \cdot {y}^{\color{blue}{\left(3 + 1\right)}}\right)\right)\right) \]

      12. pow-plus N/A

          \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left({y}^{3} \cdot y\right)}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot {y}^{3}\right)}\right)\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot {y}^{3}\right)}\right)\right)\right) \]

      15. cube-mult N/A

          \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

      19. lower-*.f64 83.4

          \[\leadsto 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right) \]

   10. Applied rewrites 83.4%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right), y\right), 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (fma y (fma (* y y) (* y (fma y (* y 0.16666666666666666) 0.5)) y) 1.0)))

C

double code(double x, double y) {
	return x * fma(y, fma((y * y), (y * fma(y, (y * 0.16666666666666666), 0.5)), y), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(y, fma(Float64(y * y), Float64(y * fma(y, Float64(y * 0.16666666666666666), 0.5)), y), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right)} \]

   2. unpow2 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right) \]

   3. associate-*l* N/A

      \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} + 1\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right), 1\right)} \]

5. Applied rewrites 91.2%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right), y\right), 1\right)} \]
6. Add Preprocessing

Alternative 5: 90.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot 0.5\right), y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 0.2) (fma x (* y y) x) (* x (* y (fma y (* y (* y 0.5)) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.2) {
		tmp = fma(x, (y * y), x);
	} else {
		tmp = x * (y * fma(y, (y * (y * 0.5)), y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.2)
		tmp = fma(x, Float64(y * y), x);
	else
		tmp = Float64(x * Float64(y * fma(y, Float64(y * Float64(y * 0.5)), y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.2], N[(x * N[(y * y), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(y * N[(y * N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.20000000000000001

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, {y}^{2}, x\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot y}, x\right) \]

      4. lower-*.f64 99.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot y}, x\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot y, x\right)} \]

   if 0.20000000000000001 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right) + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right) + 1\right) \]

      3. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)} + 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right), 1\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{2} + 1\right)}, 1\right) \]

      6. distribute-lft-in N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + y \cdot 1}, 1\right) \]

      7. *-rgt-identity N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + \color{blue}{y}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} \cdot {y}^{2}, y\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} \cdot \frac{1}{2}}, y\right), 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} \cdot \frac{1}{2}}, y\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{2}, y\right), 1\right) \]

      12. lower-*.f64 79.1

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot 0.5, y\right), 1\right) \]

   5. Applied rewrites 79.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{{y}^{4} \cdot \left(\frac{1}{2} \cdot x + \frac{x}{{y}^{2}}\right)} \]

   8. Applied rewrites 79.1%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \mathsf{fma}\left(y, y \cdot \left(y \cdot 0.5\right), y\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 93.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right), y\right), 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (fma y (fma (* y y) (* y (* (* y y) 0.16666666666666666)) y) 1.0)))

C

double code(double x, double y) {
	return x * fma(y, fma((y * y), (y * ((y * y) * 0.16666666666666666)), y), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(y, fma(Float64(y * y), Float64(y * Float64(Float64(y * y) * 0.16666666666666666)), y), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right)} \]

   2. unpow2 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right) \]

   3. associate-*l* N/A

      \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} + 1\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right), 1\right)} \]

5. Applied rewrites 91.2%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right), y\right), 1\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{6} \cdot {y}^{3}}, y\right), 1\right) \]
7. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \frac{1}{6} \cdot \color{blue}{\left(\left(y \cdot y\right) \cdot y\right)}, y\right), 1\right) \]

   2. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \frac{1}{6} \cdot \left(\color{blue}{{y}^{2}} \cdot y\right), y\right), 1\right) \]

   3. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot y}, y\right), 1\right) \]

   4. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)}, y\right), 1\right) \]

   5. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)}, y\right), 1\right) \]

   6. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}, y\right), 1\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y \cdot \left(\frac{1}{6} \cdot \color{blue}{\left(y \cdot y\right)}\right), y\right), 1\right) \]

   8. lower-*.f64 91.2

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right), y\right), 1\right) \]

8. Applied rewrites 91.2%

   \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right)\right)}, y\right), 1\right) \]

9. Final simplification 91.2%

   \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right), y\right), 1\right) \]
10. Add Preprocessing

Alternative 7: 90.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 0.2) (fma x (* y y) x) (* x (* 0.5 (* y (* y (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.2) {
		tmp = fma(x, (y * y), x);
	} else {
		tmp = x * (0.5 * (y * (y * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.2)
		tmp = fma(x, Float64(y * y), x);
	else
		tmp = Float64(x * Float64(0.5 * Float64(y * Float64(y * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.2], N[(x * N[(y * y), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(0.5 * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.20000000000000001

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, {y}^{2}, x\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot y}, x\right) \]

      4. lower-*.f64 99.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot y}, x\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot y, x\right)} \]

   if 0.20000000000000001 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right) + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right) + 1\right) \]

      3. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)} + 1\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right), 1\right)} \]

      5. +-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{2} + 1\right)}, 1\right) \]

      6. distribute-lft-in N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + y \cdot 1}, 1\right) \]

      7. *-rgt-identity N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + \color{blue}{y}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} \cdot {y}^{2}, y\right)}, 1\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} \cdot \frac{1}{2}}, y\right), 1\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} \cdot \frac{1}{2}}, y\right), 1\right) \]

      11. unpow2 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{2}, y\right), 1\right) \]

      12. lower-*.f64 79.1

          \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot 0.5, y\right), 1\right) \]

   5. Applied rewrites 79.1%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot {y}^{4}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot {y}^{\color{blue}{\left(3 + 1\right)}} \]

      3. pow-plus N/A

         \[\leadsto \left(\frac{1}{2} \cdot x\right) \cdot \color{blue}{\left({y}^{3} \cdot y\right)} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot x\right) \cdot {y}^{3}\right) \cdot y} \]

      5. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot {y}^{3}\right) \cdot y \]

      6. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {y}^{3}\right)\right)} \cdot y \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} \cdot {y}^{3}\right) \cdot y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} \cdot {y}^{3}\right) \cdot y\right)} \]

      9. associate-*l* N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({y}^{3} \cdot y\right)\right)} \]

      10. pow-plus N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{{y}^{\left(3 + 1\right)}}\right) \]

      11. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot {y}^{\color{blue}{4}}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right)} \]

      13. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot {y}^{\color{blue}{\left(3 + 1\right)}}\right) \]

      14. pow-plus N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({y}^{3} \cdot y\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot {y}^{3}\right)}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot {y}^{3}\right)}\right) \]

      17. cube-mult N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right)\right) \]

      18. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \left(y \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

      19. lower-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right)\right) \]

      20. unpow2 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right) \]

      21. lower-*.f64 79.1

          \[\leadsto x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right) \]

   8. Applied rewrites 79.1%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (fma y (* 0.16666666666666666 (* y (* y (* y (* y y))))) 1.0)))

C

double code(double x, double y) {
	return x * fma(y, (0.16666666666666666 * (y * (y * (y * (y * y))))), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(y, Float64(0.16666666666666666 * Float64(y * Float64(y * Float64(y * Float64(y * y))))), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(y * N[(0.16666666666666666 * N[(y * N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right)} \]

   2. unpow2 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right) \]

   3. associate-*l* N/A

      \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} + 1\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right), 1\right)} \]

5. Applied rewrites 91.2%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, 0.5\right), y\right), 1\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot {y}^{5}}, 1\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot {y}^{5}}, 1\right) \]

   2. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot {y}^{\color{blue}{\left(4 + 1\right)}}, 1\right) \]

   3. pow-plus N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot \color{blue}{\left({y}^{4} \cdot y\right)}, 1\right) \]

   4. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot \color{blue}{\left({y}^{4} \cdot y\right)}, 1\right) \]

   5. metadata-eval N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot \left({y}^{\color{blue}{\left(3 + 1\right)}} \cdot y\right), 1\right) \]

   6. pow-plus N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot \left(\color{blue}{\left({y}^{3} \cdot y\right)} \cdot y\right), 1\right) \]

   7. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot \left(\color{blue}{\left(y \cdot {y}^{3}\right)} \cdot y\right), 1\right) \]

   8. lower-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot \left(\color{blue}{\left(y \cdot {y}^{3}\right)} \cdot y\right), 1\right) \]

   9. cube-mult N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot \left(\left(y \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right) \cdot y\right), 1\right) \]

   10. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot \left(\left(y \cdot \left(y \cdot \color{blue}{{y}^{2}}\right)\right) \cdot y\right), 1\right) \]

   11. lower-*.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot \left(\left(y \cdot \color{blue}{\left(y \cdot {y}^{2}\right)}\right) \cdot y\right), 1\right) \]

   12. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{6} \cdot \left(\left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \cdot y\right), 1\right) \]

   13. lower-*.f64 90.9

       \[\leadsto x \cdot \mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(\left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \cdot y\right), 1\right) \]

8. Applied rewrites 90.9%

   \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{0.16666666666666666 \cdot \left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot y\right)}, 1\right) \]

9. Final simplification 90.9%

   \[\leadsto x \cdot \mathsf{fma}\left(y, 0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right), 1\right) \]
10. Add Preprocessing

Alternative 9: 90.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (fma y (fma y (* (* y y) 0.5) y) 1.0)))

C

double code(double x, double y) {
	return x * fma(y, fma(y, ((y * y) * 0.5), y), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(y, fma(y, Float64(Float64(y * y) * 0.5), y), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right) + 1\right)} \]

   2. unpow2 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right) + 1\right) \]

   3. associate-*l* N/A

      \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)} + 1\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, y \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right), 1\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{2} + 1\right)}, 1\right) \]

   6. distribute-lft-in N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + y \cdot 1}, 1\right) \]

   7. *-rgt-identity N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, y \cdot \left(\frac{1}{2} \cdot {y}^{2}\right) + \color{blue}{y}, 1\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} \cdot {y}^{2}, y\right)}, 1\right) \]

   9. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} \cdot \frac{1}{2}}, y\right), 1\right) \]

   10. lower-*.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} \cdot \frac{1}{2}}, y\right), 1\right) \]

   11. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot \frac{1}{2}, y\right), 1\right) \]

   12. lower-*.f64 88.9

       \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\left(y \cdot y\right)} \cdot 0.5, y\right), 1\right) \]

5. Applied rewrites 88.9%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)} \]
6. Add Preprocessing

Alternative 10: 88.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x \cdot \left(y \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right), x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma y (* x (* y (* y (* y 0.5)))) x))

C

double code(double x, double y) {
	return fma(y, (x * (y * (y * (y * 0.5)))), x);
}

Julia

function code(x, y)
	return fma(y, Float64(x * Float64(y * Float64(y * Float64(y * 0.5)))), x)
end

Wolfram

code[x_, y_] := N[(y * N[(x * N[(y * N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{{y}^{2} \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right) + x} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right) + x \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right)} + x \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right), x\right)} \]

5. Applied rewrites 87.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.5, y\right), x\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{fma}\left(y, x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{3}\right)}, x\right) \]
7. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \mathsf{fma}\left(y, x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right), x\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(y, x \cdot \left(\frac{1}{2} \cdot \left(y \cdot \color{blue}{{y}^{2}}\right)\right), x\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(y, x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot y\right) \cdot {y}^{2}\right)}, x\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(y, x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{2} \cdot y\right)\right)}, x\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(y, x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{2} \cdot y\right)\right), x\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(y, x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{1}{2} \cdot y\right)\right)\right)}, x\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{1}{2} \cdot y\right)\right)\right)}, x\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{2} \cdot y\right)\right)}\right), x\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(y, x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right), x\right) \]

   10. lower-*.f64 87.5

       \[\leadsto \mathsf{fma}\left(y, x \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y \cdot 0.5\right)}\right)\right), x\right) \]

8. Applied rewrites 87.5%

   \[\leadsto \mathsf{fma}\left(y, x \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot 0.5\right)\right)\right)}, x\right) \]
9. Add Preprocessing

Alternative 11: 67.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (fma y (fma y (fma y 0.16666666666666666 0.5) 1.0) 1.0)))

C

double code(double x, double y) {
	return x * fma(y, fma(y, fma(y, 0.16666666666666666, 0.5), 1.0), 1.0);
}

Julia

function code(x, y)
	return Float64(x * fma(y, fma(y, fma(y, 0.16666666666666666, 0.5), 1.0), 1.0))
end

Wolfram

code[x_, y_] := N[(x * N[(y * N[(y * N[(y * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(y \cdot 1\right)}} \]

   2. metadata-eval N/A

      \[\leadsto x \cdot e^{y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right)} \]

   3. metadata-eval N/A

      \[\leadsto x \cdot e^{y \cdot \left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto x \cdot e^{y \cdot \left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right)} \]

   5. distribute-lft-out N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)}} \]

   6. div-inv N/A

      \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right)} \]

   7. div-inv N/A

      \[\leadsto x \cdot e^{y \cdot \left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right)} \]

   8. flip-+ N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}}} \]

   9. +-inverses N/A

      \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   11. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   12. *-rgt-identity N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot 1\right)} \cdot 0}{0}} \]

   13. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

   14. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

   15. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

   16. distribute-lft-out N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

   17. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   19. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

   20. difference-of-squares N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

   21. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\color{blue}{\frac{y}{2} - \frac{y}{2}}}} \]

   22. flip-+ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y}{2} + \frac{y}{2}}} \]

   23. count-2 N/A

       \[\leadsto x \cdot e^{\color{blue}{2 \cdot \frac{y}{2}}} \]

4. Applied rewrites 71.4%

   \[\leadsto x \cdot e^{\color{blue}{y}} \]

5. Taylor expanded in y around 0

   \[\leadsto x \cdot \color{blue}{\left(1 + y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right)\right) + 1\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, 1 + y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right), 1\right)} \]

   3. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot y\right) + 1}, 1\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} + \frac{1}{6} \cdot y, 1\right)}, 1\right) \]

   5. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{1}{6} \cdot y + \frac{1}{2}}, 1\right), 1\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \]

   7. lower-fma.f64 65.4

      \[\leadsto x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, 0.16666666666666666, 0.5\right)}, 1\right), 1\right) \]

7. Applied rewrites 65.4%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \]
8. Add Preprocessing

Alternative 12: 81.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.2) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.2) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 0.2d0) then
        tmp = x
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.2) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.2:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.2)
		tmp = x;
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.2)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.2], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.20000000000000001

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{x \cdot 1} \]
   5. Step-by-step derivation

      1. *-rgt-identity 98.8

         \[\leadsto \color{blue}{x} \]

   6. Applied rewrites 98.8%

      \[\leadsto \color{blue}{x} \]

   if 0.20000000000000001 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, {y}^{2}, x\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot y}, x\right) \]

      4. lower-*.f64 66.8

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot y}, x\right) \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot y, x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot {y}^{2}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2}} \]

      2. unpow2 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot y\right)} \]

      3. lower-*.f64 66.8

         \[\leadsto x \cdot \color{blue}{\left(y \cdot y\right)} \]

   8. Applied rewrites 66.8%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.2) x (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.2) {
		tmp = x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 0.2d0) then
        tmp = x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.2) {
		tmp = x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.2:
		tmp = x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.2)
		tmp = x;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.2)
		tmp = x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.2], x, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.20000000000000001

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{x \cdot 1} \]
   5. Step-by-step derivation

      1. *-rgt-identity 98.8

         \[\leadsto \color{blue}{x} \]

   6. Applied rewrites 98.8%

      \[\leadsto \color{blue}{x} \]

   if 0.20000000000000001 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(y \cdot 1\right)}} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot e^{y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right)} \]

      3. metadata-eval N/A

         \[\leadsto x \cdot e^{y \cdot \left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto x \cdot e^{y \cdot \left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right)} \]

      5. distribute-lft-out N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)}} \]

      6. div-inv N/A

         \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right)} \]

      7. div-inv N/A

         \[\leadsto x \cdot e^{y \cdot \left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right)} \]

      8. flip-+ N/A

         \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}}} \]

      9. +-inverses N/A

         \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

      10. +-inverses N/A

          \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

      11. associate-*r/ N/A

          \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

      12. *-rgt-identity N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot 1\right)} \cdot 0}{0}} \]

      13. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

      14. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

      15. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

      16. distribute-lft-out N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

      17. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

      18. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

      19. +-inverses N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

      20. difference-of-squares N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

      21. +-inverses N/A

          \[\leadsto x \cdot e^{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\color{blue}{\frac{y}{2} - \frac{y}{2}}}} \]

      22. flip-+ N/A

          \[\leadsto x \cdot e^{\color{blue}{\frac{y}{2} + \frac{y}{2}}} \]

      23. count-2 N/A

          \[\leadsto x \cdot e^{\color{blue}{2 \cdot \frac{y}{2}}} \]

   4. Applied rewrites 47.1%

      \[\leadsto x \cdot e^{\color{blue}{y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + x \cdot y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot y + x} \]

      2. lower-fma.f64 11.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, x\right)} \]

   7. Applied rewrites 11.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, x\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} \]

      2. lower-*.f64 11.9

         \[\leadsto \color{blue}{y \cdot x} \]

   10. Applied rewrites 11.9%

       \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y \cdot y, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x (* y y) x))

C

double code(double x, double y) {
	return fma(x, (y * y), x);
}

Julia

function code(x, y)
	return fma(x, Float64(y * y), x)
end

Wolfram

code[x_, y_] := N[(x * N[(y * y), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + x \cdot {y}^{2}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot {y}^{2} + x} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, {y}^{2}, x\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot y}, x\right) \]

   4. lower-*.f64 82.6

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{y \cdot y}, x\right) \]

5. Applied rewrites 82.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y \cdot y, x\right)} \]
6. Add Preprocessing

Alternative 15: 55.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, y, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma x y x))

C

double code(double x, double y) {
	return fma(x, y, x);
}

Julia

function code(x, y)
	return fma(x, y, x)
end

Wolfram

code[x_, y_] := N[(x * y + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(y \cdot 1\right)}} \]

   2. metadata-eval N/A

      \[\leadsto x \cdot e^{y \cdot \left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right)} \]

   3. metadata-eval N/A

      \[\leadsto x \cdot e^{y \cdot \left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right)} \]

   4. metadata-eval N/A

      \[\leadsto x \cdot e^{y \cdot \left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right)} \]

   5. distribute-lft-out N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)}} \]

   6. div-inv N/A

      \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right)} \]

   7. div-inv N/A

      \[\leadsto x \cdot e^{y \cdot \left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right)} \]

   8. flip-+ N/A

      \[\leadsto x \cdot e^{y \cdot \color{blue}{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}}} \]

   9. +-inverses N/A

      \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   11. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   12. *-rgt-identity N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot 1\right)} \cdot 0}{0}} \]

   13. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

   14. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

   15. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

   16. distribute-lft-out N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

   17. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   19. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

   20. difference-of-squares N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

   21. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\color{blue}{\frac{y}{2} - \frac{y}{2}}}} \]

   22. flip-+ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y}{2} + \frac{y}{2}}} \]

   23. count-2 N/A

       \[\leadsto x \cdot e^{\color{blue}{2 \cdot \frac{y}{2}}} \]

4. Applied rewrites 71.4%

   \[\leadsto x \cdot e^{\color{blue}{y}} \]

5. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + x \cdot y} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot y + x} \]

   2. lower-fma.f64 53.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, x\right)} \]

7. Applied rewrites 53.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, x\right)} \]
8. Add Preprocessing

Alternative 16: 50.9% accurate, 111.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

4. Applied rewrites 50.0%

   \[\leadsto \color{blue}{x \cdot 1} \]
5. Step-by-step derivation

   1. *-rgt-identity 50.0

      \[\leadsto \color{blue}{x} \]

6. Applied rewrites 50.0%

   \[\leadsto \color{blue}{x} \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x \cdot {\left(e^{y}\right)}^{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (pow (exp y) y)))

C

double code(double x, double y) {
	return x * pow(exp(y), y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (exp(y) ** y)
end function

Java

public static double code(double x, double y) {
	return x * Math.pow(Math.exp(y), y);
}

Python

def code(x, y):
	return x * math.pow(math.exp(y), y)

Julia

function code(x, y)
	return Float64(x * (exp(y) ^ y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (exp(y) ^ y);
end

Wolfram

code[x_, y_] := N[(x * N[Power[N[Exp[y], $MachinePrecision], y], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024216 
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (pow (exp y) y)))

  (* x (exp (* y y))))